Integration by parts on manifold without boundary

Question

Suppose $M$ is a compact Riemannian manifold without boundary. Does the integration by part hold? For example, do we have $$ \int_M\nabla_g u\nabla_g v dx=\int_M-\Delta_g uvdx? $$ Here, $\nabla_g$ and $\Delta_g$ are defined w.r.t the Riemannian metric $g$ on the manifold.

Answer 1

Yes. $${\rm div}(u\nabla v) = u\,\triangle v + \nabla u \cdot \nabla v.$$If $M$ is compact without boundary, integrate: $$\int_M {\rm div}(u\nabla v)\,{\rm d}M =\int_M u\,\triangle v\,{\rm d}M +\int_M \nabla u\cdot \nabla v\,{\rm d}M. $$Now Stokes' theorem kills the left side, so reorganizing we get $$\int_M \nabla u \cdot \nabla v\,{\rm d}M = -\int_Mu\,\triangle v\,{\rm d}M$$as wanted.